Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.6 |

Department of Mathematics

University of California, San Diego

La Jolla, CA 92093-0112

USA

**Abstract:**

André proved that is the generating
function of all up-down permutations of even length and
is the generating function of all up-down permutation
of odd length. There are three equivalent ways to define
up-down permutations in the symmetric group . That is, a permutation
in the symmetric group is an
up-down permutation if either (i) the rise set of consists
of all the odd numbers less than , (ii) the descent set of
consists of all even number less than , or (iii) both (i) and (ii).
We consider analogues of André's results
for colored permutations of the form
where
and
under the product order.
That is, we define
if and only if
and
.
We then say a colored permutation
is
(I) an *up-not up* permutation
if the rise set of
consists
of all the odd numbers less than , (II) a *not down-down* permutation
if the descent set of
consists
of all the even numbers less than , (III) an *up-down* permutation
if both (I) and (II) hold. For ,
conditions (I), (II), and (III) are pairwise distinct. We
find -analogues of the generating functions
for up-not up, not down-down, and up-down colored permutations.

(Concerned with sequences A000111 A000182 A122045.)

Received December 19 2009;
revised version received May 5 2010.
Published in *Journal of Integer Sequences*, May 5 2010.

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